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Metric projection onto convex sets. (English. Russian original) Zbl 0578.41038
Math. Notes 31, 59-64 (1982); translation from Mat. Zametki 31, 117-126 (1982).
Let X be a real linear normed space and let $$X^*$$ be its conjugate space. For any nonempty sets M, N in X, define $$d(x,M)=\inf \{\| x- y\|:\quad y\in M\}$$ and $$d(M,N)=\sup \{d(x,N):\quad x\in M\}.$$ The set-valued mapping $$P_ Mx=\{y\in M:\quad \| x-y\| =d(x,M)\}$$ is called the metric projection from X onto M. A set M in X is called a strict convex subset, if M is convex, closed, has nonempty interior, and its boundary contains no interval. A real linear normed space X is called (RB$${\mathbb{R}})$$ space (or $$X\in (RBR))$$, if for every $$f\in X^*$$, $$\| f\| =1$$, $$\Gamma_ f=\{x:f(x)=\| x\| =1\}$$ is either empty, or singleton, or a strict convex subset in the hyperplane $$\{x\in X:f(x)=1\}$$. A metric projection $$P_ M$$ is called lower semi-continuous if for any $$x\in X$$, $$y\in P_ Mx$$ and $$x_ n\to x$$, there holds $$d(y,P_ Mx_ n)\to 0$$ is called lower H-semi-continuous if for any $$x\in X$$ and $$x_ n\to x$$ there holds $$d(P_ Mx$$, $$P_ Mx_ n)\to 0.$$
In this paper the authors study the relations of the various continuous properties of metric projection and the structure of Banach space. The main result is the following: Theorem 4. For Banach space X, the following statements are mutually equivalent: (1) $$X\in (RBR)$$; (2) for any 3-dimensional subspace $$X_ 3$$, of X, $$X_ 3\in (RBR)$$; (3) for any 3-dimensional subspace $$X_ 3$$ of X the metric projection from X onto any closed convex subset of $$X_ 3$$ is lower semi-continuous; (4) the metric projection from X onto any bounded compact convex subset $$M\subset X$$ is lower semi-continuous; (5) the metric projection from X onto any bounded compact convex subset $$M\subset X$$ lower H-semi-continuous.
Reviewer: Tingfan Xie

##### MSC:
 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 41A50 Best approximation, Chebyshev systems
##### Keywords:
metric projection; strict convex subset
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##### References:
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